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Weakly Multiplicative Distributions and Weighted Dirichlet Spaces

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Abstract

We show that if u is a compactly supported distribution on the complex plane such that, for every pair of entire functions f, g,

$$\begin{aligned} \langle u,f{\overline{g}}\rangle =\langle u,f\rangle \langle u,{\overline{g}}\rangle , \end{aligned}$$

then u is supported at a single point. As an application, we complete the classification of all weighted Dirichlet spaces on the unit disk that are de Branges–Rovnyak spaces by showing that, for such spaces, the weight is necessarily a superharmonic function.

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Acknowledgements

The authors are grateful to Trieu Le for drawing the article [2] to their attention and to Omar El-Fallah, Karim Kellay, and Vadim Ognov for helpful discussions.

Funding

Mashreghi is supported by an NSERC Discovery Grant.Ransford is supported by grants from NSERC and the Canada Research Chairs program.

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  1. Département de mathématiques et de statistique, Université Laval, Quebec City, QC, G1V 0A6, Canada

    Javad Mashreghi & Thomas Ransford

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  1. Javad Mashreghi
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  2. Thomas Ransford
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Javad Mashreghi is supported by an NSERC Discovery Grant. Thomas Ransford is supported by grants from NSERC and the Canada Research Chairs program.

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Mashreghi, J., Ransford, T. Weakly Multiplicative Distributions and Weighted Dirichlet Spaces. J Geom Anal 33, 103 (2023). https://doi.org/10.1007/s12220-022-01152-2

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